When I studied physics in junior high and high school, we always took it for granted that potential energy was equal to kinetic energy. In Lagrangian terms, $T = V$at least on average. But I realized that assumption might not always be true. I have been studying the Euler-Lagrange equation and been very confused by it. But I read in Gravity by James Hartley that Newton's law can be expressed as the Euler-Lagrange equation.
If a Lagrangian satisfies the Euler-Lagrange equation, does that mean potential energy equals kinetic energy?
In other words, suppose I have a mass held above the surface of the earth with $PE = mgh$. That's the amount of energy that upon release gradually gets converted to kinetic energy. So therefore $PE = KE$ in that case. That's what I mean. I am saying I have always assumed the system I am considering is conservative. So when I looked at the Euler Lagrange equation, it looked trivial because had (without knowing) assumed every system satisfied it. So I am trying to see if I have the story straight this time.